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| A bilinear form $A$ on a vector space $V$ (over a field $k$) is called an \emph{alternating form} if for all $v\in V$, $A(v,v)=0$. |
A bilinear form $A$ on a vector space $V$ (over a field $k$) is called an \emph{alternating form} if for all $v\in V$, $A(v,v)=0$. |
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Since for any $u,v\in V$, $$0=A(u+v,u+v)=A(u,u)+A(u,v)+A(v,u)+A(v,v)=A(u,v)+A(v,u),$$ we see that $A(u,v)=-A(v,u)$. So an alternating form is automatically a anti-symmetric, or skew symmetric form. The converse is true if the characteristic of $k$ is not $2$.
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Since for any $u,v\in V$, $$0=A(u+v,u+v)=A(u,u)+A(u,v)+A(v,u)+A(v,v)=A(u,v)+A(v,u),$$ we see that $A(u,v)=-A(v,u)$. So an alternating form is automatically a skew symmetric form. The converse is true if the characteristic of $k$ is not $2$.
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| Let $V$ be a two dimensional vector space over $k$ with an alternating form $A$. Let $\lbrace e_1,e_2\rbrace$ be a basis for $V$. The matrix associated with $A$ looks like |
Let $V$ be a two dimensional vector space over $k$ with an alternating form $A$. Let $\lbrace e_1,e_2\rbrace$ be a basis for $V$. The matrix associated with $A$ looks like |
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| \begin{center}$ |
\begin{center}$ |
| \begin{pmatrix} |
\begin{pmatrix} |
| A(e_1,e_1) & A(e_1,e_2) \\ |
A(e_1,e_1) & A(e_1,e_2) \\ |
| A(e_2,e_1) & A(e_2,e_2) |
A(e_2,e_1) & A(e_2,e_2) |
| \end{pmatrix}=r |
\end{pmatrix}=r |
| \begin{pmatrix} |
\begin{pmatrix} |
| 0 & 1 \\ |
0 & 1 \\ |
| -1 & 0 |
-1 & 0 |
| \end{pmatrix}=rS, |
\end{pmatrix}=rS, |
| $\end{center} |
$\end{center} |
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| where $r=A(e_1,e_2)$. The skew symmetric matrix $S$ has the property that its diagonal entries are all $0$. $S$ is called the $2\times 2$ \emph{alternating} or \emph{symplectic matrix}. |
where $r=A(e_1,e_2)$. The skew symmetric matrix $S$ has the property that its diagonal entries are all $0$. $S$ is called the $2\times 2$ \emph{alternating} or \emph{symplectic matrix}. |
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| $A$ is called \emph{non-singular} or \emph{non-degenerate} if there exist a vectors $u,v\in V$ such that $A(u,v)\neq 0$. $u,v$ are necessarily non-zero. Note that the associated matrix $rS$ is non-singular iff $r\neq 0$ iff $A$ is non-singular. |
$A$ is called \emph{non-singular} or \emph{non-degenerate} if there exist a vectors $u,v\in V$ such that $A(u,v)\neq 0$. $u,v$ are necessarily non-zero. Note that the associated matrix $rS$ is non-singular iff $r\neq 0$ iff $A$ is non-singular. |
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| In the two dimensional vector space case above, if $A$ is non-singular, we can re-scale the basis elements so that $r=1$. This means that the matrix associated with $A$ is the alternating matrix. A two-dimensional vector space which carries a non-singular alternating form is sometimes called an \emph{alternating} or \emph{symplectic hyperbolic plane}. Some authors also call it simply a hyperbolic plane. But here on PlanetMath, we will reserve the shorter name for its cousin in the category of quadratic spaces. Let's denote an alternating hyperbolic plane by $\mathcal{A}$. |
In the two dimensional vector space case above, if $A$ is non-singular, we can re-scale the basis elements so that $r=1$. This means that the matrix associated with $A$ is the alternating matrix. A two-dimensional vector space which carries a non-singular alternating form is sometimes called an \emph{alternating} or \emph{symplectic hyperbolic plane}. Some authors also call it simply a hyperbolic plane. But here on PlanetMath, we will reserve the shorter name for its cousin in the category of quadratic spaces. Let's denote an alternating hyperbolic plane by $\mathcal{A}$. |
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| \textbf{Remark.} In general, it can be shown that if $V$ is an $n$-dimensional vector space equipped with a non-singular alternating form $A$, then $V$ can be written as an orthogonal direct sum of the alternating hyperbolic planes $\mathcal{A}$. In other words, the associated matrix for $A$ has the block form |
\textbf{Remark.} In general, it can be shown that if $V$ is an $n$-dimensional vector space equipped with a non-singular alternating form $A$, then $V$ can be written as an orthogonal direct sum of the alternating hyperbolic planes $\mathcal{A}$. In other words, the associated matrix for $A$ has the block form |
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| \begin{center}$ |
\begin{center}$ |
| \begin{pmatrix} |
\begin{pmatrix} |
| S & \boldsymbol{0} & \cdots & \boldsymbol{0} \\ |
S & \boldsymbol{0} & \cdots & \boldsymbol{0} \\ |
| \boldsymbol{0} & S & \cdots & \boldsymbol{0} \\ |
\boldsymbol{0} & S & \cdots & \boldsymbol{0} \\ |
| \vdots & \vdots & \ddots & \vdots \\ |
\vdots & \vdots & \ddots & \vdots \\ |
| \boldsymbol{0} & \boldsymbol{0} & \cdots & S \\ |
\boldsymbol{0} & \boldsymbol{0} & \cdots & S \\ |
| \end{pmatrix},\mbox{ where }\boldsymbol{0}= |
\end{pmatrix},\mbox{ where }\boldsymbol{0}= |
| \begin{pmatrix} |
\begin{pmatrix} |
| 0 & 0 \\ 0 & 0 |
0 & 0 \\ 0 & 0 |
| \end{pmatrix}. |
\end{pmatrix}. |
| $\end{center} |
$\end{center} |
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| Furthermore, $n$ is even. $V$ is called a symplectic vector space. |
Furthermore, $n$ is even. $V$ is called a symplectic vector space. |
